Remarks on Physics as Number Theory (original) (raw)

An interface between physics and number theory

Journal of Physics: Conference Series, 2011

We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides a mathematical route from an algebraic description of non-relativistic, non-field theoretic quantum statistical mechanics to one of relativistic quantum field theory. Such a description necessarily involves treating the algebra of polyzeta functions, extensions of the Riemann Zeta function, since these occur naturally in pQFT. This provides a link between physics, algebra and number theory. As a by-product of this approach, we are led to indicate inter alia a basis for concluding that the Euler gamma constant γ may be rational.

Number Theory and Physics at the Crossroads

2011

This is the fifth gathering of this series of workshops in the interface of number theory and string theory. The first one was held at the Fields Institute in 2001. The subsequent workshops were all held at BIRS, and this one was the fourth BIRS workshop. The workshop met from May 8th to May 13th for five days. Altogether 40 mathematicians (number theorists and geometers) and physicists (string theorists) converged at BIRS for the five-days' scientific endeavors. About half of the participants were familiar faces, while the other half were new participants. There were 24 one-hour talks, and our typical working day started at 9:30am and ended at 9:30pm. Indeed we worked very hard, and the workshop was a huge success! This fourth workshop was partially dedicated to Don Zagier on the occasion of his completing the first life cycle and reaching the 60 years of age. This series of workshops has been getting very prominent in the community and attracted more applications than the workshop could accommodate. There was overwhelming urge from the participants to organize a five-day workshop in two or three years time at BIRS. We are indeed planning to submit a proposal for the next BIRS workshop in due course.

Toward the Unification of Physics and Number Theory

World Scientific Reports in Advances in Physical Science , 2019

In Part I, we introduce the notion of simplex-integers and show how, in contrast to digital numbers, they are the most powerful numerical symbols that implicitly express the information of an integer and its set theoretic substructure. In Part II, we introduce a geometric analogue to the primality test. Our geometric form provokes a novel hypothesis about the distribution of prime-simplexes that, if solved, may lead to a proof of the Riemann hypothesis. Specifically, if a geometric algorithm predicting the number of prime simplexes within any bound n-simplexes or associated An lattices is discovered, a deep understanding of the error factor of the prime number theorem would be realized – the error factor corresponding to the distribution of the non-trivial zeta zeros. In Part III, we discuss the mysterious link between physics and the Riemann hypothesis. We suggest how quantum gravity and particle physicists might benefit from a simplex-integer based quasicrystal code formalism. An argument is put forth that the unifying idea between number theory and physics is code theory, where reality is information theoretic and 3-simplex integers form physically realistic aperiodic dynamic patterns from which space, time and particles emerge from the evolution of the code syntax. Finally, an appendix provides an overview of the conceptual framework of emergence theory, an approach to unification physics based on the quasicrystalline spin network. 7/26/18 Correction: In Section 3.2 on page 13, it says: Conjecture 2 : if p is composite, let p = ab, where a≠1, b≠1 then at least one of the coefficients is not divisible by p It should say: Conjecture 2 : if p is composite, let p=a b, where a and b are prime and a≠1, b≠1 then at least one of the coefficients is not divisible by p.

The Algebraic Structure of Physical Quantities

Journal of Mathematical Chemistry, 2000

We present a general algebraic basis for arbitrary systems of units such as those used in physical sciences, engineering, and economics. Physical quantities are represented as q-numbers: an ordered pair u = {u, label u }, that is, u ∈ q = X × W B. The algebraic structure of the infinite sets of labels that represent the "units" has been established: such sets W B are infinite Abelian multiplicative groups with a finite basis. W B is solvable as it admits a tower of Abelian subgroups. Extensions to include the possibility of rational powers of labels have been included, as well as the addition of named labels. Named labels are an essential feature of all practical systems of units. Furthermore, q is an Abelian multiplicative group, and it is not a ring. q admits decomposition into one-dimensional normed vector spaces over the field X among members with equivalent labels. These properties lead naturally to the concept of well-posed relations, and to Buckingham's theorem of dimensional analysis. Finally, a connection is made with a Group Ring structure and an interpretation in terms of the observable properties of physicochemical systems is given.

Historical Changes in the Concepts of Number, Mathematics and Number Theory

This essay traces the history of three interconnected strands: changes in the concept of number; in the nature and importance of arithmetike (҆ αριθμητικη), the study of the qualities of number, which evolved into number theory; and in the nature of mathematics itself, from early Greek mathematics to the 20 th century.

PHYSICAL POSSIBILITY AND DETERMINATE NUMBER THEORY

It's currently fashionable to take Putnamian modeltheoretic worries seriously for mathematics, but not for discussions of ordinary physical objects and the sciences. However, I will argue that (under certain mild assumptions) merely securing determinate reference to physical possibility suffices to rule out the kind of nonstandard interpretations of our number talk Putnam invokes. So, anyone who accepts determinate reference to physical possibility should not reject determinate reference to the natural numbers on Putnamian model-theoretic grounds. 1

Application of Algebraic Number Theory to Rational Theory

International Journal of Advanced Research in Science, Communication and Technology, 2023

This paper contained some notations connected with algebraic number theory and indicates some of its applications in the Gaussian field namely K(i) = (−1)..

Fundamental Perceptions in Contemporary Number Theory

Nova Science Publishers, 2023

The current state and future directions of numerous facets of contemporary number theory are examined in this book ”Fundamental Perceptions in Contemporary Number Theory” from a unified standpoint. The theoretical foundations of contemporary theories are unveiled as a consequence of simple challenges. Additionally, this book makes an effort to present the contents as simply as possible. It is primarily intended for novice mathematicians who have tried reading other works but have struggled to comprehend them due to complex reasoning.

Numbers as Properties (Accepted Manuscript)

Synthese, 2023

Although number sentences are ostensibly simple, familiar, and applicable, the justification for our arithmetical beliefs has been considered mysterious by the philosophical tradition. In this paper, I argue that such a mystery is due to a preconception of two realities, one mathematical and one nonmathematical, which are alien to each other. My proposal shows that the theory of numbers as properties entails a homogeneous domain in which arithmetical and nonmathematical truth occur. As a result, the possibility of arithmetical knowledge is simply a consequence of the possibility of ordinary knowledge 1 .

On Measurements, Numbers and p-Adic Mathematical Physics

arXiv (Cornell University), 2012

In this short paper I consider relation between measurements, numbers and p-adic mathematical physics. p-Adic numbers are not result of measurements, but nevertheless they play significant role in description of some systems and phenomena. We illustrate their ability for applications referring to some sectors of p-adic mathematical physics and related topics, in particular, to string theory and the genetic code.